Example: Lvl. 1 Shuckle, Contrary, Brave nature, No IVs or EVs in Speed used Rock-polish. Would Rock-polish Fail if it got to the point where it would lower the speed to less than 1? If not would it effect Gyro-ball?

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No. If given just a decimal for a stat value, a stat automatically rounds up to 1, even if it's something like .0001. Same thing happens when your attack is so weak, it wouldn't do anything. (Imagine a level 1 magby using ember on a level 100 Omastar. The damage would do next to nothing, but it rounds up to 1.)